Questions tagged [orlicz-spaces]

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Orlicz-Sobolev Spaces

let $A$ an N-function and suppose that $$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty $$ we denote by $\widehat{A}$ an N-function equal to A near infinity and $\widehat{A}$ ...
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65 views

Lyapounov's inequality for Orlicz norms

When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$. This inequality is a special (or rather limit) case of ...
6
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242 views

Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$

Let $\Phi$ be a Youngs's function, i.e. $$ \Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$$ for some $\varphi$ satifying $\varphi:[0,\infty)\to[0,\infty]$ is increasing $\varphi$ is lower semi ...
3
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1answer
124 views

Hoeffding to bound Orlicz norm

I have been reading from Weak Convergence and Empirical Processes, and came across the following: Let $a_1,\ldots,a_n$ be constants and $\epsilon_1,\ldots,\epsilon_n\sim$Rademacher. Then $\mathbb{P}...
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1answer
75 views

Reference Request: $L^p(x)$/(Musielak–Orlicz space) analogue of classical $L^p$ result

Fix a non-empty open domain $\Omega\subseteq \mathbb{R}^d$ with compact closure, and a finite Borel measure $\mu$ on its closure $\overline{\Omega}$. In Halmos' book it is shown that: Classical ...
3
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1answer
114 views

Which Orlicz functions $f$ make the function $f^{-1}\left(\frac{\sum_{j=1}^s f(x_j)}{s}\right)$ convex?

Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be an Orlicz function, or sometimes referred to as an Young function, i.e. it is a convex, non-decreasing function such that $f(0)=0$. I am trying to study the ...
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55 views

Concavification of Orlicz function

In the Handbook of the Geometry of Banach Spaces, vol 1, page 855, Johnson and Schechtman say that if $M$ is an Orlicz function, the following are equivalent: The unit vector basis of the Orlicz ...
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1answer
96 views

Definition of an Orlicz modular space

In Nowak (1989), a modular $\rho$ on a vector lattice is defined by the following properties (N1) $\rho(x)=0\implies x=0$; (N2) $\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$; (N3) ...
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Modular which is metrizing but does not satisfy the $\Delta_2$ condition

Let $\Phi$ be a nice Young function (N-function) and $(\Omega,\mathcal{F},P)$ a probability space such that either $P$ is diffuse on a set of non-zero probability or $P$ is purely atomic and there are ...
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1answer
220 views

Young’s complement of $ x \mapsto x \, {\log^{+}}(x) $, $ N $-functions and Orlicz spaces

The function $ \Phi: \mathbb{R} \to \mathbb{R} $ is an $ N $-function if and only if it is continuous, even and convex with: $ \displaystyle \lim_{x \to 0} \frac{\Phi(x)}{x} = 0 $. $ \displaystyle \...
3
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1answer
295 views

Looking for a reference for a version of the “The reverse Lebesgue dominated convergence theorem” for the Orlicz spaces

Please I need a reference where I can find a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces analogous to Theorem 1.2.7 in Semilinear Elliptic Equations for ...
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80 views

Boudedness of linear operator between generalized Orlicz spaces

I am using the notations, definitions, and results of the Section X of [1] on generalized Orlicz spaces. We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is ...